Problem 60 - Introductory Algebra for College Students

Question

Factor each polynomial using the greatest common binomial factor. $$x(x+3)-8(x+3)$$

Step-by-Step Solution

Answer

The given polynomial \(x(x+3)-8(x+3)\) factors to \((x+3)(x-8)\) when using the greatest common binomial factor.

Step 1: Identify the binomial factor

The first step in factoring the given expression is to identify the greatest common binomial factor between the two terms, which in this case is \(x+3\). This binomial is present in both terms \(x(x+3)\) and \(-8(x+3)\).

Step 2: Factor out the Binomial

Next, the common binomial factor \((x+3)\) is factored out from the original expression. This is accomplished by reaffirming each part of the expression as a product of the binomial factor and its remaining factor. Therefore, \(x(x+3)-8(x+3)\) can be rewritten as \((x+3)(x-8)\).

Step 3: Check the Factorisation

The effectiveness of factorization can be verified by expanding the factored form back to its original polynomial. When \((x+3)(x-8)\) is expanded, it returns to the original polynomial \(x(x+3)-8(x+3)\). This confirms that the factorisation process was correct.